From the result (3.75) one may obtain a number of identities obeyed by the binomial coefficients. For example, we may decide not to distinguish between colors 1 and 2; i.e. a ball of either color is declared to have color ‘a’. Then from (3.75) we must have, on the one hand,

从结果（3.75）的二项式系数我们可以得到多个等式。例如，我们可能决定不区分颜色1和2;例如,任选一个球,称其颜色为'a'。然后从（3.75）我们必然有，从一个方面看，

with

且

But the event $r_a$ can occur for any values of $r_1, r_2$ satisfying (3.77), and so we must have also, on the other hand,

对任何$r_1，r_2$满足（3.77）的值,事件$r_a$都可能发生，所以我们从另外一方面看必有，

Then, comparing (3.76) and (3.78), we have the identity

那么,比较(3.76)和(3.78),得到等式

Continuing in this way, we can derive a multitude of more complicated identities obeyed by the binomial coefficients. For example,

如此继续,我们可以根据二项式系数推出多个更复杂的等式。 例如，

In many cases, probabilistic reasoning is a powerful tool for deriving purely mathematical results; more examples of this are given by Feller (1950, Chap. 2 & 3) and in later chapters of the present work.

在许多情况下，概率推理是获得纯粹数学结果的有力工具; Feller（1950，第2章和第3章）以及本书后面的章节给出了更多这方面的例子。